Sains Ma1aysiana 26(1): 69-77 (1997)                                                                           Sains Fizis dan Gunaan/

                                                                                                                                         Physical and Applied Sciences

Laguerre Polynomials and Functions

Karim B. Yaacob

Faculty of Science and Art

Universiti Putra Malaysia Terengganu

21030 Kuala Terengganu Terengganu Malaysia

ABSTRAK

Mengikut teori persamaan perbezaan hipergeometrik konfluen, polinomial Laguerre Lⁿ_m(x), memenuhi persamaan xy" + (m+l-x)y' + (n-m)y=0, untuk kesemua nombor genap positif n ≥ m, termasuk sifar. Di dalam keadaan n bukan nombor genap, penyelesaian kepada persamaan Laguerre bukan berbentuk polinomial, tetapi suatu siri tak terhingga yang sukar untuk disimpulkan di dalam bentuk fungsi tertutup. Suatu kaedah baru dikenali di mana jadual polinomial Laguerre (n nombor genap) dan fungsi Laguerre (n bukan nombor genap) dapat diterbitkan melalui penyelesaian kepada persamaan kamiran. Kaedah ini menggunakan teknik novel melalui penyelesaian kepada persamaan Bateman dan persamaan Yaacob-Bateman untuk mendapat penyelesaian kepada persamaan jejari Schroedinger untuk atom hidrogen.

ABSTRACT

In the theory of the confluent hypergeometric differential equation, Laguerre polynomials Lⁿ_m(x), satisfy the equation xy" + (m+l-x)y' + (n-m)y=0, for non-negative integers of n ≥ m, including zero. When n are non-integers, the solutions to the equation are not polynomials, but infinite series that is difficult to evaluate in closed from. A new method is introduced whereby it is possible to generate a table of Laguerre polynomials (for integer n) and Laguerre functions (for non-integer n) through evaluation of one common integral equation. The method uses the Bateman and the Yaacob-Bateman equations as new tools of solution to the radial Schroedinger equation for the hydrogen atom.

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